While trying to think about possible interesting notions of algebraic independance over a skew field, I am wondering where in mathematics appears the notion of being independent, or free over something.

I am currently thinking of

**(1)** Linear independence in linear algebra, for elements of a $R$-module,

**(2)** Algebraic independence for elements of a $K$-algebra (over a field $K$),  on which there is a natural action of $K[x_1,\dots,x_n]$,

**(3)** Free groups, presentation by generators and relations.

**(4)** Independence in probability.

**(5)** Shelah's notion of abstract independence for types in model theory, particularly in stable theories.

<blockquote><b>Q1.</b> Any other natural/interesting examples ? Or comments about these ones ?</blockquote>

Whereas I clearly see a connexion between the first 3 items : the question of the existence of a non trivial 'equation' of a certain kind bounding the elements, item (4) seems of a different nature (at least I do not understand its nature).

<blockquote><b>Q2.</b> Are there analogies or more between these items ? Is there someting that the probabilistic notion of independence shares with say item (1) ? What is a good notion of independance and why ? Why is it usefull ?</blockquote>

These questions are very naïve, so naïve answers are allowed  !